Remarks on diameter 2 properties

Abstract

If $X$ is an infinite-dimensional uniform algebra, if $X$ has the Daugavet property or if $X$ is a proper $M$-embedded space, every relatively weakly open subset of the unit ball of the Banach space $X$ is known to have diameter 2, i.e., $X$ has the diameter 2 property. We prove that in these three cases even every finite convex combination of relatively weakly open subsets of the unit ball have diameter 2. Further, we identify new examples of spaces with the diameter 2 property outside the formerly known cases; in particular we observe that forming ${\ell }_p$-sums of diameter 2 spaces does not ruin diameter 2 structure.

To appear in Journal of Convex Analysis

Cited by in corpus (10)

• On strongly norm attaining Lipschitz maps
• The Daugavet property in the Musielak-Orlicz spaces
• Stability of average roughness, octahedrality, and strong diameter 2 properties of Banach spaces with respect to absolute sums
• Relatively weakly open convex combinations of slices
• Almost squareness and strong diameter two property in tensor product spaces
• Almost square and octahedral norms in tensor products of Banach spaces
• Locally octahedral and locally almost square Köthe-Bochner spaces
• An improvement of a theorem of Heinrich, Mankiewicz, Sims, and Yost
• On octahedraliy and Müntz spaces
• Strong diameter two property and convex combination of slices reaching the unit sphere